Subgroups — #44553275

Order 32128.

704 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Dic₂₅₁	C₂	1	2	yes	maximal
2	#30631586	C₂	1	2	yes	maximal, complemented
3	#30631562	C₂	1	4	yes	maximal, complemented
4	#30631553	C₂	1	1	yes	maximal, characteristic, complemented
5	#30631547	C₂	1	2	yes	maximal, complemented
6	C₄×C₄⋊Dic₂₅₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₂₅₁⋊#95	C₂	1	2	yes	maximal, complemented
8	C₂×C₄×D₄×C₂₅₁	C₂	1	1	yes	maximal, characteristic
9	C₂³×Dic₂₅₁	C₂²	1	4	yes	—
10	D₄×Dic₂₅₁	—	2	8	no	—
11	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₂₅₁	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₂₅₁	C₂²	1	4	yes	—
14	C₂×C₂²⋊Dic₂₅₁	C₂²	1	4	yes	—
15	C₂×C₄×Dic₂₅₁	C₂²	1	2	yes	—
16	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
17	C₂₅₁⋊#95	C₂²	1	2	yes	—
18	C₂₅₁⋊#95	C₂²	1	4	yes	—
19	C₂₅₁⋊#95	C₂²	1	1	yes	characteristic
20	C₂₅₁⋊#95	C₂²	1	2	yes	—
21	C₂²×D₄×C₂₅₁	C₂²	1	1	yes	characteristic
22	C₂³×C₄×C₂₅₁	C₂²	1	2	yes	—
23	C₄×D₄×C₂₅₁	C₄	1	8	yes	complemented
24	C₂×C₄⋊C₄×C₂₅₁	C₂²	1	1	yes	characteristic
25	C₂×C₂²⋊C₄×C₂₅₁	C₂²	1	2	yes	—
26	C₂×C₄²×C₂₅₁	C₂²	1	1	yes	characteristic
27	C₂²×Dic₂₅₁	—	2	8	no	—
28	C₂²×Dic₂₅₁	—	2	8	no	—
29	C₂²×Dic₂₅₁	—	4	4	no	—
30	C₂²×Dic₂₅₁	C₂³	1	4	yes	—
31	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
32	C₂²×Dic₂₅₁	C₂³	1	2	yes	—
33	C₄⋊Dic₂₅₁	—	2	4	no	—
34	C₄⋊Dic₂₅₁	—	2	4	no	—
35	C₄⋊Dic₂₅₁	—	4	4	no	—
36	C₂²⋊Dic₂₅₁	—	2	8	no	—
37	C₄×Dic₂₅₁	—	2	4	no	—
38	C₂⁴×C₂₅₁	C₂³	1	2	yes	—
39	C₂×D₄×C₂₅₁	C₂×C₄	1	4	yes	—
40	C₂×D₄×C₂₅₁	D₄	1	4	yes	—
41	C₂×D₄×C₂₅₁	Q₈	1	4	yes	—
42	C₂²×C₄×C₂₅₁	—	4	4	no	—
43	C₂²×C₄×C₂₅₁	C₂×C₄	1	8	yes	complemented
44	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
46	C₂²×C₄×C₂₅₁	C₂³	1	2	yes	—
47	C₂²×C₄×C₂₅₁	C₂³	1	1	yes	characteristic
48	C₄⋊C₄×C₂₅₁	C₂×C₄	1	4	yes	complemented
49	C₂²⋊C₄×C₂₅₁	C₂×C₄	1	8	yes	complemented
50	C₄²×C₂₅₁	C₂×C₄	1	4	yes	complemented
51	C₂×Dic₂₅₁	—	2	2	no	—
52	C₂×Dic₂₅₁	—	2	2	no	—
53	C₂×Dic₂₅₁	—	2	2	no	—
54	C₂×Dic₂₅₁	—	2	4	no	—
55	C₂×Dic₂₅₁	—	2	2	no	—
56	C₂×Dic₂₅₁	—	2	4	no	—
57	C₂×Dic₂₅₁	—	2	2	no	—
58	C₂×Dic₂₅₁	—	2	4	no	—
59	C₂×Dic₂₅₁	—	4	16	no	—
60	C₂×Dic₂₅₁	#41	1	4	yes	—
61	C₂³×C₂₅₁	—	2	4	no	—
62	C₂³×C₂₅₁	—	2	4	no	—
63	C₂³×C₂₅₁	C₂²×C₄	1	4	yes	—
64	C₂³×C₂₅₁	C₂×D₄	1	4	yes	—
65	C₂³×C₂₅₁	C₂×Q₈	1	4	yes	—
66	C₂³×C₂₅₁	C₂⁴	1	1	yes	characteristic
67	D₄×C₂₅₁	C₄⋊C₄	1	16	yes	complemented
68	C₂×C₄×C₂₅₁	—	2	2	no	—
69	C₂×C₄×C₂₅₁	—	2	2	no	—
70	C₂×C₄×C₂₅₁	—	2	4	no	—
71	C₂×C₄×C₂₅₁	—	4	8	no	—
72	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
73	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
74	C₂×C₄×C₂₅₁	C₂²×C₄	1	4	yes	—
75	C₂×C₄×C₂₅₁	C₂²×C₄	1	2	yes	—
76	C₂×C₄×C₂₅₁	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₂₅₁	C₂×Q₈	1	2	yes	—
78	C₂²×C₂₅₁	—	2	4	no	—
79	C₂²×C₂₅₁	—	2	4	no	—
80	C₂²×C₂₅₁	—	2	8	no	—
81	C₂²×C₂₅₁	—	2	8	no	—
82	C₂²×C₂₅₁	C₂×C₄⋊C₄	1	8	yes	—
83	C₂²×C₂₅₁	C₂³×C₄	1	1	yes	characteristic
84	C₂²×C₂₅₁	C₂²×D₄	1	1	yes	characteristic
85	C₂²×C₂₅₁	C₂²×Q₈	1	1	yes	characteristic
86	C₂²×C₂₅₁	C₂×#41	1	2	yes	—
87	C₂²×C₂₅₁	H(5,𝔽₂)	1	2	yes	—
88	C₄×C₂₅₁	—	2	2	no	—
89	C₄×C₂₅₁	—	2	2	no	—
90	C₄×C₂₅₁	—	2	4	no	—
91	C₄×C₂₅₁	C₂×C₄⋊C₄	1	4	yes	—
92	Dic₂₅₁	—	2	4	no	—
93	Dic₂₅₁	—	4	2	no	—
94	Dic₂₅₁	—	4	4	no	—
95	C₂×C₂₅₁	—	2	8	no	—
96	C₂×C₂₅₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
97	C₂×C₂₅₁	C₄×#41	1	1	yes	characteristic
98	C₂×C₂₅₁	#518	1	1	yes	characteristic
99	C₂×C₂₅₁	#546	1	2	yes	—
100	C₂×C₂₅₁	#554	1	2	yes	—
101	C₂₅₁	#2349	1	1	yes	characteristic, complemented, Sylow
102	#2349	—	251	1	no	maximal, Sylow
103	C₂×C₄×D₄	—	251	2	no	—
104	C₂×C₄×D₄	D₂₅₁	1	1	yes	characteristic
105	#389	—	251	2	no	—
106	#385	—	251	4	no	—
107	#382	—	251	1	no	—
108	#380	—	251	2	no	—
109	C₄×C₄⋊C₄	—	251	1	no	—
110	C₂×#95	—	251	2	no	—
111	C₂²×D₄	C₂×D₂₅₁	1	1	yes	characteristic
112	C₂³×C₄	—	251	4	no	—
113	C₂³×C₄	C₂×D₂₅₁	1	2	yes	—
114	C₄×D₄	—	502	8	no	—
115	C₄×D₄	Dic₂₅₁	1	8	yes	complemented
116	C₂×C₄⋊C₄	—	251	4	no	—
117	C₂×C₄⋊C₄	—	251	2	no	—
118	C₂×C₄⋊C₄	—	251	2	no	—
119	C₂×C₄⋊C₄	C₂×D₂₅₁	1	1	yes	characteristic
120	C₂×C₂²⋊C₄	—	251	4	no	—
121	C₂×C₂²⋊C₄	C₂×D₂₅₁	1	2	yes	—
122	C₂×C₄²	—	251	2	no	—
123	C₂×C₄²	C₂×D₂₅₁	1	1	yes	characteristic
124	#95	—	251	1	no	—
125	#95	—	251	2	no	—
126	#95	—	251	4	no	—
127	#95	—	251	1	no	—
128	#95	—	251	2	no	—
129	C₂⁴	C₂²×D₂₅₁	1	2	yes	—
130	C₂×D₄	C₂×Dic₂₅₁	1	4	yes	—
131	C₂×D₄	D₁₀₀₄	1	4	yes	—
132	C₂×D₄	Dic₅₀₂	1	4	yes	—
133	C₂²×C₄	—	4	4	no	—
134	C₂²×C₄	—	1004	4	no	—
135	C₂²×C₄	—	502	8	no	—
136	C₂²×C₄	—	502	8	no	—
137	C₂²×C₄	—	251	4	no	—
138	C₂²×C₄	—	251	2	no	—
139	C₂²×C₄	—	251	2	no	—
140	C₂²×C₄	C₂×Dic₂₅₁	1	8	yes	complemented
141	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
142	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
143	C₂²×C₄	C₂²×D₂₅₁	1	2	yes	—
144	C₂²×C₄	C₂²×D₂₅₁	1	1	yes	characteristic
145	C₄⋊C₄	—	1004	4	no	—
146	C₄⋊C₄	—	502	4	no	—
147	C₄⋊C₄	—	502	4	no	—
148	C₄⋊C₄	C₂×Dic₂₅₁	1	4	yes	complemented
149	C₂²⋊C₄	—	502	8	no	—
150	C₂²⋊C₄	C₂×Dic₂₅₁	1	8	yes	complemented
151	C₄²	—	502	4	no	—
152	C₄²	C₂×Dic₂₅₁	1	4	yes	complemented
153	C₂³	—	2	4	no	—
154	C₂³	—	2	4	no	—
155	C₂³	C₂²×Dic₂₅₁	1	4	yes	—
156	C₂³	C₂×D₁₀₀₄	1	4	yes	—
157	C₂³	C₂×Dic₅₀₂	1	4	yes	—
158	C₂³	C₂³×D₂₅₁	1	1	yes	characteristic
159	D₄	C₄⋊Dic₂₅₁	1	16	yes	complemented
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	4	8	no	—
164	C₂×C₄	—	1004	16	no	—
165	C₂×C₄	—	502	2	no	—
166	C₂×C₄	—	502	4	no	—
167	C₂×C₄	—	502	2	no	—
168	C₂×C₄	—	502	4	no	—
169	C₂×C₄	—	502	2	no	—
170	C₂×C₄	—	502	4	no	—
171	C₂×C₄	—	502	2	no	—
172	C₂×C₄	—	502	2	no	—
173	C₂×C₄	—	251	4	no	—
174	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
175	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
176	C₂×C₄	C₂²×Dic₂₅₁	1	4	yes	—
177	C₂×C₄	C₂²×Dic₂₅₁	1	2	yes	—
178	C₂×C₄	C₂×D₁₀₀₄	1	2	yes	—
179	C₂×C₄	C₂×Dic₅₀₂	1	2	yes	—
180	C₂²	—	2	8	no	—
181	C₂²	—	2	8	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	C₂×C₄⋊Dic₂₅₁	1	8	yes	—
185	C₂²	C₂³×Dic₂₅₁	1	1	yes	characteristic
186	C₂²	C₂²×D₁₀₀₄	1	1	yes	characteristic
187	C₂²	C₂²×Dic₅₀₂	1	1	yes	characteristic
188	C₂²	#41×D₂₅₁	1	2	yes	—
189	C₂²	#22548760	1	2	yes	—
190	C₄	—	2	2	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	4	no	—
193	C₄	—	1004	4	no	—
194	C₄	—	1004	2	no	—
195	C₄	—	502	4	no	—
196	C₄	C₂×C₄⋊Dic₂₅₁	1	4	yes	—
197	C₂	—	2	8	no	—
198	C₂	C₂²×C₄⋊Dic₂₅₁	1	1	yes	characteristic
199	C₂	#41×Dic₂₅₁	1	1	yes	characteristic
200	C₂	#30632162	1	1	yes	characteristic
201	C₂	#30632332	1	2	yes	—
202	C₂	#30632396	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₂₅₁
commutator	C₂²×C₂₅₁	C₂³×C₄
Fitting	C₂×C₄×D₄×C₂₅₁	C₂
socle, cosocle	C₂³×C₂₅₁	C₂⁴
prime radical	C₂²×D₄×C₂₅₁	C₂²
hypercenter	C₂×C₄×D₄	D₂₅₁
hypocenter	C₂₅₁	#2349
LCS	G ▷ C₂²×C₂₅₁ ▷ C₂₅₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
2. #30631586	C₂	semidirect
3. #30631562	C₂	semidirect
4. #30631553	C₂	semidirect
5. #30631547	C₂	semidirect
6. C₄×C₄⋊Dic₂₅₁	C₂	semidirect
7. C₂×C₂₅₁⋊#95	C₂	semidirect
23. C₄×D₄×C₂₅₁	C₄	semidirect
43. C₂²×C₄×C₂₅₁	C₂×C₄	semidirect
48. C₄⋊C₄×C₂₅₁	C₂×C₄	semidirect
49. C₂²⋊C₄×C₂₅₁	C₂×C₄	semidirect
50. C₄²×C₂₅₁	C₂×C₄	semidirect
67. D₄×C₂₅₁	C₄⋊C₄	semidirect
101. C₂₅₁	#2349	semidirect
115. C₄×D₄	Dic₂₅₁	semidirect
140. C₂²×C₄	C₂×Dic₂₅₁	semidirect
148. C₄⋊C₄	C₂×Dic₂₅₁	semidirect
150. C₂²⋊C₄	C₂×Dic₂₅₁	semidirect
152. C₄²	C₂×Dic₂₅₁	semidirect
159. D₄	C₄⋊Dic₂₅₁	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×D₄×Dic₂₅₁	2
#30631586	2
#30631562	2
#30631553	2
#30631547	2
C₄×C₄⋊Dic₂₅₁	2
C₂×C₂₅₁⋊#95	2
C₂×C₄×D₄×C₂₅₁	2
#2349	251